Materials, Interfaces, and Electrochemical Phenomena

A New Technique for Predicting the Shape of Solution-Grown Organic Crystals

Daniel Winn and Michael F. Doherty Dept. of Chemical Engineering, University of Massachusetts, Amherst, MA 01003

A method for predicting the shape of solution-grown organic crystals is presented. The shapes depend on properties of their internal crystal structure, as well as on the processing en®ironment. The model de®eloped here can account for the effects of sol- ®ent and may be extended to account for the influence of certain types of additi®es. This method may be used as a first approach for including crystal shape in the o®erall design and optimization of organic solids᎐ processes. The technique has been used successfully to predict the shape of adipic acid grown from water, biphenyl from toluene, and ibuprofen from polar and nonpolar sol®ents.

Introduction The shape of crystalline organic solids is an important fac- actions between organic molecules in the solid state has led tor in the design and operation of solid᎐liquid separation sys- to a burgeoning of ideas of how these forces influence mor- tems. Crystal morphology affects the efficiency of down- phology. Saska and MyersonŽ. 1983 and Berkowitch-Yellen stream processesŽ. such as filtering, washing, and drying , and Ž.1985 were among the first to calculate solid-state energies influences material properties such as bulk density and me- of organic materials for the purpose of morphological model- chanical strength which play a major role in storage and han- ing. An especially important contribution has been the work dling. It also affects particle flowability, agglomeration, and of Roberts and co-workersŽ Docherty and Roberts, 1988a; mixing characteristics, as well as their redissolution proper- Clydesdale et al., 1991. who developed the program HABIT, ties. Since many organic solids produced in the chemical in- a fast and easy tool for calculating the interaction energy dustry are sold as end-products, the quality and efficacy of within organic crystals. these materials also depend in large part on crystal shape. The results of this research are models and methods that The focus in recent years on specialty chemical processes, provide estimates of the relative growth rates of various crys- like pharmaceuticals, has added to the demand for methods tal faces solely from knowledge of the internal crystal struc- that can predict and control crystal morphologyŽ Davey, 1991; ture. Since the construction of the growing crystal polyhe- Tanguy and Marchal, 1996. . dron depends only on relative rates of growthŽ Hoffman and Since the early days of crystal shape modeling, the primary Cahn, 1972. , the crystal shape can be determined. These pre- focus has been the influence of the internal crystal structure dictions are often accurate for vapor grown crystals, but are on the external crystal habit. BravaisŽ. 1866 , Friedel Ž. 1907 , generally not accurate for solution growth. They fail to ac- Donnay and HarkerŽ. 1937 , as well as Hartman and Perdok count for the influence of the crystallizer solution and the Ž.1955 and others Ž see the review by Myerson and Gind, 1993 . , driving forceŽ. supersaturation that are widely thought to be all proposed similar relationships: the greater the surface major factors affecting shapeŽ Wells, 1946; Myerson and Gind, density of molecules on a face, the stronger the lateral inter- 1993. . actions among molecules, and the more stable and slow grow- Of the many attempts to account for these factors, the most ing it must be. Thus, low Miller index faces are dense, slow quantitative success has been achieved by Bennema and co- growing, and dominate the crystal shape, as seen experimen- workersŽ. Liu and Bennema, 1993, 1996; Liu et al., 1995 . They tally. More recent work in this field has focused on the actual have applied the methods of statistical mechanics to examine magnitude of interactions within the crystal. The advent of solvent and supersaturation effects at the crystal-solution in- force-fields to calculate van der Waals and electrostatic inter- terface, and have combined these formulations with detailed growth kineticsᎏthe well known theories of Burton, Cabrera Correspondence concerning this article should be addressed to M. F. Doherty. and FrankŽ. 1951 ᎏas a means of predicting crystal shape

AIChE Journal November 1998 Vol. 44, No. 11 2501 from solution. Their model requires solvent-crystal physical properties that are estimated from detailed experiments or molecular dynamics simulations. This new technique, how- ever, has not yet been widely applied. The need for lengthy simulations or experiments greatly limits its utility for process engineering applications. We propose a method for predicting crystal shape that can account for realistic processing conditions, and does not ne- cessitate extensive molecular simulation. It is based on simi- lar physical principles to those proposed by Bennema and co-workersŽ. Liu et al., 1995; Liu and Bennema, 1996 , but requires only knowledge of pure component properties that, for many systems, are readily available: the crystal structure and internal energy of the solid and the pure component sur- Figure 1. Circular 2-D nucleus on a crystal face. face free energy of the solvent. Solid᎐solvent interface prop- erties are estimated using a classical approach that we de- scribe later in the article. ⌬ The simplicity and ease of the method permits its use dur- critical size, G reaches a maximum; the critical radius at ing the discovery stage of an organic solids processŽ such as which this occurs is given by to guide experiments. . In addition, it may be advantageous to ␥ edge include this shape calculation, along with the traditional pre- VM r s Ž.3 dictions for crystal size and size distribution in the overall c ⌬␮ flowsheet design and optimization of these processes.

A nucleus of a size equal to rc is in unstable equilibrium: Thermodynamics of Crystal Surfaces smaller ones redissolve; larger ones grow. The change in free energy associated with the creation of one nucleus of critical In a solution crystallizer with seeds or nuclei, a supersatu- size is ration causes the transfer of solute from solution onto the surfaces of individual crystals. With the addition of molecules 2 ␲␥edge Vd onto a macroscopically flat crystal surfaceŽ. a crystal face , ⌬ s Ž.M Gc Ž.4 there is a change in the Gibbs free energyŽ.Ž kJrmol at fixed ⌬␮ T and P. of the entire system corresponding to Note that when the supersaturation is high, the critical size ⌬Gsy N⌬␮q␥ A Ž.1 can be very small, down to a minimum of one molecule. Similar equations can be developed for nuclei of other ⌬␮s␮y␮ where sat, the difference between the chemical shapes. For example, with a square nucleus of dimensions l potentials of the solute in solution and in the crystal phase by l, Eq. 3 and Eq. 4 become Ž.in units of energy per mole , N is the number of moles ␥ r 2 ␥ edge transferred, is the specific surface free energyŽ erg cm. , 2 VM l s 5 and A is the area of the new surface that is formed. The first c ⌬␮ Ž. term represents a decrease in free energy due to a phase change, and the second term represents an increase in free ␥ edge 2 4Ž.VdM ⌬G s 6 energy due to surface formation. The new surface is an edge c ⌬␮ Ž. surface, a surface of monolayer height wrapping around a molecule, or cluster of molecules. To indicate this, we will The general formula for a regular n-sided nucleus is write ␥ as ␥ edge. Equation 1 can be used to examine the free energy change ␥ edge VM associated with one individual cluster of molecules, termed a l s2tan Ž.␲rn Ž.7 two-dimensionalŽ. 2-D nucleus. To a first approximation, the c ⌬␮ ␥ edge 2-D nucleus is a disc with an isotropic , a volume of 2 2 ␥ edge ␲ r d and an edge area of 2␲ rd, where d is the interlayer Ž.VdM ⌬G s n tan ␲rn 8 spacingŽ. A˚ of the given face Ž see Figure 1 . . Since N is the c Ž.⌬␮ Ž. 3r ⌬ volume divided by the molar volume VM Žcm mol. G is a function of r as given byŽ unit conversion factors are omitted The free energy change can be expressed in units per mole of for clarity. critical nuclei by multiplying Eq. 8 by Avogadro’s number. It can then be simplified by defining an edge free energy per ⌬␮ ⌬Gsy␲ rd2edgeq2␲ rd␥ Ž.2 mole of edge molecules VM ␾ edge s␥ edge edge AM Ž.9 At low supersaturations and ␥ edge )0, the free energy edge 2r change is positive and increases with r. However, at a certain where AM is the areaŽ cm mol of edge molecules. . Since

2502 November 1998 Vol. 44, No. 11 AIChE Journal edge 2 s Ž AMAM. NV d, the free energy per mole of critical nuclei Evidence from crystal growth simulationsŽ Gilmer and Ben- can be expressed as nema, 1972.Ž , as well as from experiments Hunan et al., 1981; Jetten et al., 1984. , indicate that rough faces grow an order Ž.␾ edge 2 of magnitude faster than the slowest flat faces on a crystal. ⌬G s n tan ␲rn 10 c Ž.⌬␮ Ž. The growth of flat faces is thought to occur by the move- ment of monolayer steps. Material is added to steps that propagate laterally and spread out over a face. Each com- This equation will be used later in the article as part of mod- plete layer results in the advancement of the face by a dis- els for predicting the growth rates of crystal faces. tance of monolayer height in the direction of the surface nor- The concept described aboveᎏthe existence of a maxi- mal. Very convincing evidence of this mechanism can be seen mum in the free energy change as a 2-D nucleus increases in in recent in-situ atomic force microscopyŽ. AFM studies of sizeᎏis applicable primarily to nuclei on low index crystal organic crystal growthŽ Manne et al., 1993; Carter et al., 1994; faces. Since the operating strategy in many crystallization Yip and Ward, 1996. . processes is to maintain a relatively constant supersaturation Growth by steps requires the formation of a stepped inter- within the metastable zone, the existence of a physically face, as well as the desolvation, transport, and incorporation meaningful maximumŽ. larger than one molecule depends on of material into the step. One possible source of steps are ␥ edge being an adequately large, positive value. Nuclei on low 2-D nuclei; the edge of the nucleus creates a step perpendic- index faces have, in general, a large edge free energy due to a ular to the face. Several authorsŽ Lewis, 1974; Ohara and large loss of lateral solid᎐solid intermolecular interactions at Reid, 1973; Chernov, 1984; van der Eerden, 1993; Markov, the edgesŽ. an increase in internal energy at the edges . On 1995. present detailed descriptions of growth by 2-D nucle- high index faces, however, there are usually fewer and weaker ation. The most realistic of these is the so-called birth and lateral interactions, and ␥ edge can be very small. Under real- spread modelŽ see the figure on p. 37 of Ohara and Reid, istic supersaturations, the above equations could yield a criti- 1973.Ž. . In Markov’s 1995 development of 2-D nucleation from cal radius that is smaller than one molecule. The physical solution, the expression for the relative growth rate of faces interpretation is that the existence of individual molecules has a proportional dependence on two functions: dispersed on these faces lowers the free energy of the system. expŽ.Ž.y⌬G r3RT , and exp yUrRT , where U is the desolva- Such faces tend to roughen, losing their distinct orientation, c tion activation energyŽ. kJrmol . As discussed in the Ap- and develop an interfacial region which is a mixture of solid pendix, it can be shown that faces with the large ⌬G also and ambient-phase molecules. If the ambient phase is a sol- c have a small U, and vice versa; they vary inversely, but not vent, especially one that is chemically similar to the solid, it proportionally. The velocity of a face decreases with increas- can contribute to this phenomena by interacting with edges ing ⌬G , but not by an exponential dependence. We, there- in a fashion which lowers ␥ edge.w If the ambient phase is c fore, propose as an approximation that the relative growth chemically dissimilar to the solidŽ such as polar solventrnon- rate follows an inverse proportionality dependence, and, thus, polar solid. , then it might have the opposite effect. It may the expression increase edge free energies, and favor the formation of flat faces.x 1 There is another major influence favoring this surface R A 11 rel ⌬ Ž. roughening phenomena. Both Burton et al.Ž. 1951 and Jack- Gc sonŽ. 1958 showed that the formation of more than one nu- cleus leads to cooperative effects, and that there can be a The range of applicability is the same as in the original model, substantial increase in the configurational entropy of the sys- ⌬ ) ⌬ that is, Gcc3RT. When G approaches 3RT, the face tem when multiple nuclei are dispersed on a face. In effect, becomes roughened by the presence of many nuclei, and edge ␥ is not just a function of internal energyŽ the unsatu- growth no longer proceeds by a stable, flat growth mecha- rated interactions of nuclei. , but also depends on the num- nismŽ. van der Eerden, 1993 . ber, size, and configuration of nuclei. If the internal energy ⌬ 4 ␥ edge When Gc 3RT, as is the case when is large and effects are small enough, it is possible for the configurational ⌬␮ is small, growth by a 2-D nucleation mechanism is ex- effects to dominate, and even with no supersaturation, a re- tremely slow. Burton, Cabrera and Frankwx BCFŽ. 1951 , in edge arrangement to a rough interface can be favorable Ž␥ is their seminal article, suggested that there must be another effectively zero. . This is generally referred to as thermody- source of steps if growth was to proceed under these condi- namic roughening. In the case where configurational effects tions. They suggested that screw dislocations on crystal faces edge do not dominate internal energy effects Ž␥ is a small, pos- provide an infinite source of steps. The edge from a screw itive value. , then the face is rough only above some critical dislocation grows laterally and rotates about itself, resulting supersaturationŽ. Bennema, 1993 . This is termed kinetic in a growth spiral of stepsŽ. see Figure 2 . Once a moving step roughening. has reached the initial position of the directly underlying step, the spiral has made a complete turn. This rotation increases Growth Kinetics the height of the spiral, causing the macroscopic face to ad- For faces that are rough, there are no energetic barriers vance. The BCF expression for the rate of growth of a face is associated with incorporating material into the crystal sur- thus face. Thus, growth in these directions is diffusion andror d= ® heat-transfer limited. Since these processes are relatively s Rrel Ž.12 isotropic, rough faces tend to lose their distinct orientation. y

AIChE Journal November 1998 Vol. 44, No. 11 2503 when ␾ kink ) RT; and Eq. 11, the 2-D nucleation model, which might occur when ␾ kink - RT. We cannot be certain of 2-D nucleation when ␾ kink - RT, since it is possible that spi- ral growth occurs without any kink integration limitations Ž.that is, by mass-transfer limited spiral growth . A compari- son between these two possibilities requires calculating abso- lute growth rates, which are a function of supersaturation as well as transport and molecular propertiesŽ such as vibra- tional frequencies. . Since one cannot, in practice, calculate these absolute growth rates, we will make the assumption that ␾ kink - RT results in 2-D nucleated growth. Under this low kink energy regime, we can also apply the condition that faces ⌬ - r with Gc 3RT are rough and grow by a fast diffusion lim- Figure 2. Square spiral originating from a screw dislo- ited mechanism. Thus, we distinguish three growth regimes, cation on a crystal face. all of which are a function of the face-dependent parameters ␾ edge and ␾ kink. In a later section we describe methods for estimating their values. where d is the step height, ® is the lateral velocity of the step, and y is the lateral distance between steps. In the limit- Spiral geometry ing case whereŽ. 1 we ignore bulk or surface diffusion effects The geometry of a growth spiral originates in the work of ŽŽ..Ž.as argued by Chernov 1984 for solution growth , and 2 we BCF with further developments by Cabrera and Levine assume that steric barriers are isotropic, then the face depen- Ž.1956 . An excellent summary of their work can be found in dent portion of the step velocity depends mainly on the den- the review of Ohara and ReidŽ. 1973 . In these early develop- sity of kink sites Žvacancies in the step where material can ments, the spiral is thought to be circular with numerous incorporate. multiplied by the distance the step is propa- kinks, and its edge velocity is assumed to be surface-diffu- gated sion-limited. KaichewŽ. 1962 and Voronkov Ž. 1973 examined ᎏ y the geometry of spirals with straight edges and few kinks a ®A q ␾ kinkr 1 r ᎏ ap 1 0.5 exp Ž.RT Ž.13 low temperature high kink energy regime where the edge velocity is limited by kink integration kinetics. Muller- Ž. where ap is the distanceŽ. A˚ that edge is propagated by adding Krumbhaar 1978 performed Monte Carlo simulations of a monolayer to it; the term in brackets is the probability of spiral crystal growth under both regimes, with results that are finding a kink in a step at equilibrium, and ␾ kink is the work, consistent with these original theoretical developments. Since or free energy change, to create a kink along an edgeŽ Burton the model developed above for solution growth assumes only et al., 1951. . Therefore, with only face dependent parame- kink integration kinetics, we will focus on the physics of ters, Eq. 12 becomes faceted spirals formed from completely straightŽ or approxi- mately straight. edges. d y1 The basic principle governing spiral motion is that the very R A a 1q0.5 exp Ž.␾ kinkrRT Ž.14 rel y p top of a rotating spiral is stationary. Since a 2-D layer that neither grows nor dissolves is of critical size, the very top ®s This expression remains valid only when ␾ kink is greater than edge of the spiral must be of critical length. Thus, 0 for F RT Ž.Burton et al., 1951 . When it is smaller, kinks are suffi- l lcc. The velocity of edges greater than l depends on the ciently numerous that edges are roughened, and edge free growth mechanism; for our case, velocity depends on the kink energy is sufficiently low that 2-D nucleation occurs. density. In Eq. 13, we presented an expression for the equi- Kink free energy can also be defined as librium kink density, and it is not a function of the edge length. Thus, we would expect that any edge on a spiral with kink kink kink ␾ s␥ A Ž.15 a length greater than lc will travel at its steady-state velocity. VoronkovŽ. 1973 has presented a nonequilibrium derivation where ␥ kink is the free energy of forming a kink per unit kink of kink density, and has also determined that kink density is area, and Akink is the kink areaŽ cm2rmol. . not a strong function of edge length. It is, therefore, reason- edge The general formula for the interstep spacing is yA␬␥ , able to assume that all edges greater than lc have the same where ␬ depends on the geometry of the spiral. In the next density of kinks. The edge velocity can be expressed as section, we derive this relationship and give explicit values ␬ ␾ edge s edge edge = ®s0, lFl for . In addition, since AMMp, and since A a c s s ®s® ) Ž.17 VM constant, Eq. 14 reduces to Ž.eq , l lc

d y1 ®Ž. R A 1q0.5 exp Ž.␾ kinkrRT Ž.16 where eq is just the velocity of the straight edge with its rel ␬␾edge equilibrium concentration of kinks as given by Eq. 13. One must also consider which, and how many, straight We now have two expressions for the relative rates of edges are likely to appear on a growth spiral. Since the maxi- growth of crystal faces: Eq. 16, the screw dislocation model, mum rotational symmetry in a 2-D lattice is six, there cannot

2504 November 1998 Vol. 44, No. 11 AIChE Journal be an n-sided spiral, with n)6, such that all edges have the pal fast direction j that is perpendicular to the stable edge ) ® 4 ® same molecular density and bonding. All n-sided n 6 spi- Ž.ji, the interstep spacing becomes rals have all of the most densely packed edges with high kink A ␥ edge energies and slow growth, and additional edges that are less yi 4 Ž.19 dense and faster growing. It is reasonable to assume that only the most dense, slowest growing edges dominate the spiral The six-sided spiral can be constructed in the same manner shape, and this reduces the number of spiral facets to a num- as the Kaichew constructionŽ. Voronkov, 1973 . In this case, ber no greater than six. As we will discuss in detail in a later the initial edge moves forward six times during a complete ␲r section, the most dense edges dominating spiral shape are rotation. Each movement is a distance lc sinŽ. 2 6 . Substitut- those that contain first nearest neighbor interactions. ing lc from Eq. 7, the interstep distance is proportional to We can determine the number of possible first nearest 6␥␬edge Ž.s6 . If there are differences in the velocities of the neighbor interactions by examining the packing arrangement three directions, the interstep distance can be calculated in a of molecules in two dimensions. Only five lattices are possi- manner analogous to Eq. 18 ble: square, rectangular, centered rectangular, oblique, and hexagonalŽ. Giacovazzo, 1992 . It is trivial to show that for all 22 y A 2qq ␥ edge 20 these arrangements, it is only possible to have one, two or i ®r® ® r® Ž. ž/ji ki three edge directions with first nearest neighbors. When a face contains two edges with first nearest neighbors, the spi- where ® and ® are the two faster growing edges. Ž jk ral is four-sided. It can be exactly square if it stems from The pre-factors of ␥ edge in Eqs. 18 through 20 are the ␬ square packing; it can be thought of as quasi-square if it stems values necessary for application of the spiral growth model. from oblique packing. . When there is only one edge with first nearest neighbors, this slow growing edge is much larger in size than the other fast growing, less dense edgesᎏthe spiral Estimating Physical Properties is very elongated and can be thought of as rectangular. If The development thus far presents ␾ edge and ␾ kink as the there are three edges with first nearest neighbors, then the key solvent-dependent physical properties affecting crystal spiral is six-sided. Thus, under our first nearest neighborrkink shape. Their values can be derived from ␥ edge and ␥ kink. The integration limited assumptions, there exists only these lim- former is an interfacial property in the classical sense; mi- ited spiral shapes. croscopy has shown that edges on crystal faces delineate a KaichewŽ. 1962 employed the principles of Eq. 17 to esti- distinct boundary between the crystal and ambient phases. mate the interstep distance in the case of a square spiral: a On the the other hand, because a kink is of molecular dimen- four-sided spiral with identical bonding structure in the two sions it is not an interface on a macroscopic scale. Neverthe- edge directions, and thus the same values of ® and ␥ edge in less, in order to develop a first-order estimation of ␥ kink we both edge directions. In this construction, growth begins from assume that the free energy to form a kink can be approxi- an initial edge dislocation. The entire edge moves forward a mated by an effective interfacial free energy for the forma- distance infinitesimally greater than lc, so as to create a new tion of a 2-D interface separating the solvent and solid at edge, perpendicular to it, of length infinitesimally greater than kink sites. lc. This new edge can now grow at its steady-state velocity, There are several potential approaches for obtaining val- and it repeats the same movement. Four such motions are ues of interfacial thermodynamic quantities: experimental required to form a complete spiral. During this time, the ini- measurements, molecular level simulations, group contribu- tial edge has moved forward a total of 4lc, and, thus, the tion estimations, and so on. The first approach is not feasi- distance between steps is 4lcc. Given l from Eq. 5, the inter- ble, because it is not possible to measure interfacial free en- step distance is proportional to 8␥␬edge Ž.s8. ergy at molecular-sized interfaces. Molecular modeling at the We might also consider the case where, due to asymmetry organic solid-solution interface is feasible, and has recently in the molecules and chemical interactions, the two dense been employed for morphological modeling purposesŽ Boek edges have small differences in their energetic properties. The et al., 1994. . The main drawback of this approach is the large result is a situation where ␥ edge and ␾ kink vary only slightly computational time required to simulate the solution in con- in the two edge directions, while the velocity varies more sub- tact with many edges on many faces. For purposes of process stantially because of the exponential dependence in Eq. 13. engineering, this method does not yet seem efficient; how- Using subscripts i and j to denote the two edge directions, i ever, with advances in computing and in computational being the slower of the two, the spiral construction yields an chemistry, it may soon become the desirable approach. A interstep distance given by group contribution method would also be extremely useful, although no precise methods exist for organic interfacial sys- 4 tems. y A 4q ␥ edge 18 i ®r® Ž. We, therefore, employ a simple classical approach for esti- ž/ji mating interfacial properties, one first suggested in the work of Hildebrand on the thermodynamics of mixingŽ see Hilde- ®G® ␥ edge where ji, and where we have assumed that re- brand and Scott, 1962. , and later fully developed by Girifalco mains essentially independent of edge direction. and GoodŽ. 1957 . The formation of an interface at equilib-

Equation 18 allows us to make an estimate of yi in the rium between two phases is thought of as a three-stage pro- case of only one stable, slow growing edge on a face. When cess: creating a surface between the first phase and a vac- the fast growth directions can be approximated by one princi- uum, creating a surface between the second phase and a vac-

AIChE Journal November 1998 Vol. 44, No. 11 2505 uum, then bringing the two surfaces into contact. The first while the electrostatic contributions are mostly limited to a two steps result in a free energy increase, often termed the few specific interactions, usually hydrogen bonds. Thus, Eq. free energy of cohesion Ž.for each surface . The third step 23 can be applied to all kink interfaces, except those that are causes a free energy decrease which is often termed the free formed by these specific electrostatic bonds. For these kinks, energy of adhesion. The development of Girifalco and Good we can approximate the interfacial free energy with experi- suggests that the free energy of adhesion per unit area can be mental measurements of liquid᎐liquid interfacial free ener- approximated by twice the geometric mean of the two cohe- gies of substances of a similar chemical natureŽ available in sive surface free energies. Thus, the net free energy per area the literature. . A common situation is when the solvent is of an interface between phase ␣ and phase ␤ can be written highly polarŽ. such as water , and a crystal kink surface is as dominated by unsaturated hydrogen bondsŽ such as carboxyl groups. . For such cases, ␥ kink should be quite small. Faces ␥ ␣ , ␤␣␤s␥ q␥ y2'␥␥ ␣␤ Ž.21 with these kinks should have high kink densities and fast growth, and the overall crystal morphology will not be sensi- tive to this approximation. This representation is often referred to as the geometric Another restriction on the application of the geometric mean approximation. Its physical premise is analogous to that mean approximation to surface free energy is that it is most of the regular solution model of mixing, and, thus, it is most accurate when the intermolecular distance at the interface valid when the surface enthalpy can be approximated by the Ž.that is, normal to the interface is close in magnitude to the surface energy, and the net change in entropy for interface intermolecular distances of the two individual phasesŽ Giri- s formation is zeroŽ. or T 0 . It also holds approximately true falco and Good, 1957. . It is often the case in organic crystal- if the change in entropy at the interface also follows the geo- lization that the solute has a larger molecular volume than metric mean rule. Equation 21 tends to match experimental the solvent; thus, we expect that crystal edges formed from ᎐ ␥ ␣ results for solution solution interfacial free energy, where the most dense chains of moleculesŽ smallest intermolecular ␥ ␤ and are taken from measured surface tensions in contact distances. are those that are closest in molecular dimension with airŽ. Barton, 1983 . to the solvent. These are the edges to which Eqs. 22 and 23 Equation 21 can be further extended to the case where one can be applied. The other less dense edges generally do not of the phases has some electrostatic or polar component to appear on the surface structures, and, thus, their exact inter- w its surface free energy. We define electrostatic component of face properties are not needed for morphological prediction. the free energy in the way normally prescribed in solubility Their large intermolecular spacing results in weak solid᎐solid parameter theory for internal energy; that is, some portion of bondingŽ. low kink energies and high kink densities , and also the total free energy stems from dispersive forces, and the reduces steric barriers to mutual dissolutionŽ the edge may ᎏ ᎏ rest the electrostatic component is due to dipole-dipole be ‘‘porous’’with respect to the solvent. . These edges are dis- r and or hydrogen bonding. The division of free energy in this ordered and fast growing with disordering analogous to sur- way is not on as firm a theoretical footing as when it is ap- face roughening discussed earlier. plied to internal energy only. However, because surface free Although the theoretical aspects of edge roughening have energy tends to correlate with the bulk internal energyŽ Bar- been discussed by several authorsŽ see the review by Chernov ton, 1983. , it is suspected that surface entropy does correlate and NishinagaŽ.. 1987 , the effect cannot be captured quanti- with the amount and type of forces contributing to the inter- tatively without a means for predicting kink energies on the x nal energy. If we assume that there are no induced electro- less dense edges. However, some insight can be gathered from static interactions in the free energy of adhesion, then Eq. 21 molecular simulations of ideal Kossel, cubic packed crystals can be rewritten as Ž.Muller-Krumbhaar, 1978; Markov, 1995 . These studies have been able to successfully predict physical phenomenaŽ such ␣ ␤␣␤ ␣X ␤ . ␥ , s␥ q␥ y2'␥␥ Ž.22 as face roughening and spiral geometry when assuming that only first nearest neighbor interactions affect surface struc- ture. All longer-distance interactions are assumed a priori to where the prime symbol denotes only the dispersiveŽ van der result in rough, unstable, and fast growing edges. We pro- Waals. component of the surface free energy. pose using this same assumption to determine the likely sta- Writing this relationship for the effective interfacial free ble edges within solution grown crystals. A strict definition of energy at a kink nearest neighbor is ‘‘a molecule whose center of mass is within the first coordination sphere of a central molecule’s.’’ For X crystals with mainly dispersive forces between molecules, this ␥ kinks␥ crystq␥ solvy2 ␥␥ cryst solv Ž.23 ii' i coordination sphere can be defined by the material’s charac- teristic packing diameter where the superscripts denote crystal and solvent phase sur- r face free energies, and the subscript i denotes a kink site on 3 1 3 D s2 V Ž.24 a particular edge on a crystal face. mmž/4␲ In writing Eq. 23, we have allowed the solvent to be polar ᎐ 3 or nonpolar, while we have explicitly restricted the solid so- where Vm is the molecular volumeŽ. A˚ . Any chain of lid interactions to dispersive forces only. Thus, its range of molecules within the crystal whose intermolecular distance li applicability is somewhat limited. However, the forces within Ž.A˚ is within a distance Dm is considered to have nearest organic crystals are often dominated by dispersive forces, neighbor interactions. We can state this criterion as

2506 November 1998 Vol. 44, No. 11 AIChE Journal F kink limD Ž.25 edge has a very low ␾ , the rough growth front is not likely to have a ␾ kink significantly greater than zero. We can roughly Edges that satisfy the criterion are thought to be reasonably estimate the edge free energy as the average of the one cal- stable, have similar bonding distances to those in the solvent, culated ␾ kink, and zero and, thus, have kink interfaces to which Eq. 23 can be ap- plied. 1 ␾ edges ␾ kink Ž.29 For crystals that contain chains of hydrogen bonds, the 2 packing motif is a function of the dispersive forces between these chains. Therefore, the diameter characteristic of dis- In the case where the one stable edge has a large kink free s maxr␲ 0.5 max persive forces is defined by Dmhkl2Ž A . , where A hklenergy, the edge is unroughened, while other edge directions is the reticular density of the most dense face intersected by are assumed roughened. We expect that spiral growth occurs, the chains. For crystals that contain 2-D layers of hydrogen and that the spiral is bounded by the one stable edge and a bonds, the characteristic diameter is the distance between ad- roughened growth front that is approximately perpendicular jacent layers. Some of the intermolecular interactions in a to it. Because this growth front has bonding and packing ge- ) crystal that do not satisfy the criterionŽ. that is, limD may ometry that is not vastly different from the stable edge, we nevertheless have dispersive energies greater than or equal to suspect that its kink free energy is as close to the stable edge’s those that doŽ. a result of the molecular asymmetry . We must value as possible; that is, it has the largest possible ␾ kink that assume that such interactions are first nearest neighbors as leads to roughening. As mentioned earlier, the theoretical well. We will also include hydrogen bonds as first nearest value for this limit is RT, which is approximately 2.5 kJrmol neighbors, since they are by definition short-range interac- at room temperature. Thus, the edge free energy can be esti- tions between adjacent molecules. mated from the average of the calculated ␾ kink and RT The remaining unknown parameter in this morphological model is edge free energy. Liu and BennemaŽ. 1996 have 1 ␾ edges Ž.␾ kink q RT Ž.30 shown that on a square nucleus, the kink surface in one edge 2 direction is the edge surface of the other direction. Thus, ␥ edge s␥ kink jiwhere i and j are the two edge directions. Since we assume in the 2-D nucleation model that the nuclei are © isotropicᎏthat they are not the Wulff shape, but a cluster Sol ent properties that is dominated by the relatively isotropic packing and Solvent surface free energies have been measured for many bonding properties in the solidᎏthen we can define an aver- solvents in contact with airŽ. Adamson, 1990; Kaelble, 1971 . age ␥ edge for the nucleus. Therefore, on a given face In addition, within certain classes of solvents, ␥ solv correlates with their bulk solubility parameters.w See BartonŽ. 1983 for ␥ edgef␥ kink Ž.26 details on solubility parameters, their values, and their corre- lations to surface properties.x Therefore, unreported solvent ␥ kink␥ kink surface free energies, as well as the dispersive components of where is the mean of all i on a given face. This average value can be used in the spiral growth formalism, the surface free energy, can be estimated using these rela- since the rate expression is a much stronger function of edge tionships. Ž. velocity than of edge free energy. A further approximation The empirical correlations in Kaelble 1971 and Barton Ž. which simplifies the calculation for a square nucleus is 1983 have the form

X ␥ solvf0.143␦ 2Ny1r3V 1r3 Ž.31 ␾ edgef␾ kink Ž.27 dA M where ␦ is the dispersive solubility parameter, N is Avo- A kink surface does not correspond to an edge surface on dA gadro’s number, and VM is the molar volume. For example, a six-sided nucleus or spiral. Hexagonal packing results in ␥ solv ᎐ water at room temperature has a measured of 72.8 edges that have two unsaturated solid solid contacts. Thus, r 2 ␦ s r 3 1r2 erg cm . Using d 7.6Ž cal cm. Ž Hansen and Beer- we approximate the edge free energy by Xsolv bower, 1971. and Eq. 31, we find that ␥ makes up about 10 ergrcm2 of the total. ␾ edgef2␾ kink Ž.28 Crystal properties A difficulty in estimating edge free energy arises when Eq. 25 yields only one first nearest neighbor edge direction on a We assume that changes in internal energy dominate the face. We have assumed that all other edge directions on the creation of crystal surfaces in contact with a vacuum. Thus, face are rough. In the case where the ‘‘stable’’ edge itself has the surface free energy is approximately equal to the surface a low kink free energyŽ. less than RT , all edge directions are energy, which can be estimated from intermolecular forces rough and the 2-D nucleation mechanism is appliedŽ see the Ž.Kitaigorodsky, 1973 . The calculation of internal energy growth kinetics section. . The nucleus can be pictured as a within crystals and at crystal surfaces has been widely applied relatively isotropic disk dominated by two rough growth using the well known attachment energy techniqueŽ Saska and fronts: one in the direction of the ‘‘stable’’edge, and the other Myerson, 1983; Berkovitch-Yellin, 1985; Docherty and which we assume is approximately perpendicular to it.Ž The Roberts, 1988b; Green et al., 1993. , and there are excellent nucleus is quasi-square with rough edges. . Since the ‘‘stable’’ software programs available, most notably HABITŽ Clydes-

AIChE Journal November 1998 Vol. 44, No. 11 2507 dale et al., 1991. and Cerius2 . These programs are fast and Ž.5 If the largest ␾ kink F RT, apply the 2-D nucleation easy to use, and require only the crystal structure and choice model. For each likely face: ⅷ ⌬ of force field as input. Calculate Gc using Eq. 10. The output from these calculations provides a per mole ⅷ Calculate the relative growth rate by Eq. 11. For those ⌬ F value of the intermolecular energy between a central molecule faces that are rough due to Gc 3RT, the growth rate can and every other molecule in the crystal. These be estimated as an order of magnitudeŽ. ten times faster than molecule᎐molecule interactions are traditionally referred to the average of the unroughened face growth rates. as ‘‘bond energies’’Ž. noncovalent , and have a corresponding Ž.6 If the largest ␾ kink G RT, apply the screw dislocation bond distance between centers of mass. Because of the re- model. For each likely face: peating nature of the crystal lattice, these bonds form chains ⅷ Calculate relative growth rate with Eq. 16. Use largest ␾ kink ␬ running throughout the crystal. If the chain is parallel to a i on the face, and the corresponding from Eqs. 18 face, it can form the boundary of an edge on that face. De- through 20Ž that is, single edge, square spiral, or hexagonal ␾ cryst noting bond energy along an edge i as i , bond length by spiral. . kink s r liimi, and an effective kink area of A V l , we can calcu- From the relative growth rates, a shape can be drawn using late the effective surface energy at a kink by the Wulff constructionŽ. Hoffmann and Cahn, 1972 . Note that if the flat, slow growing faces intersect only two crystallo- ␥ cryst s␾ crystr kink graphic axes, there is rough growth in the direction of the iiiA Ž.32 third axis, and the resulting crystal is an elongated needle shape. If the slow growing faces intersect only one principal Note that we only perform this calculation on bond chains axis, then the crystal is a thin flat plate. that satisfy the criterion of Eq. 25, and with only the disper- sive component of the energy. Example: Adipic acid in water Likely Crystal Faces The shape of adipic acid crystals has been studied exten- sively with particular interest in improving the acid’s flowabil- The law of Bravais, Freidel, Donnay and HarkerŽ. BFDH ityŽ. Klug and van Mil, 1994 and its role in pill formation of Žsee the computer implementation of this law, MORANG, by pharmaceutical compoundsŽ. Grant et al., 1991 . It is usually Docherty and RobertsŽ.. 1988a is an efficient means of crystallized from aqueous solution into a flat plate with large choosing the likely crystal faces for which one should calcu- Ä4100 faces bordered by small Ä4Ä4Ä4 011 , 111 , 002 , and Ä4 102 late face growth rates. It states that the larger a face’s inter- faces. Electron micrographs of adipic acid grown from aque- planar spacing Ž.d , the slower its growth and the greater hkl ous solutions in a fluidized-bed crystallizer just below a satu- its size. This corresponds with the kinetic models we have ration temperature of 303 K are presented in Figure 3. Ex- presentedᎏfaces with larger d have higher densities of hkl perimental details are given in WinnŽ. in press . molecules, and, therefore, more nearest neighbors, more sta- Attachment energy and BFDH calculations have been pre- ble edges, larger ␥ edge, and slower growth. viously performed by Davey et al.Ž. 1992 , and we have re- It is possible for several crystallographic forms with differ- ent interplanar spacings to have identical first nearest neigh- bor bonding structuresᎏthey have identical bond chains defining their edges. Of these forms, we would expect those with larger dhkl to have greater edge surface area, and, thus, slightly larger edge and kink energies. However, our method of estimating these properties is not sensitive to step height. To compensate, we propose choosing the likely faces by the BFDH approach, but for any set of forms with identical sta- ble bond chains, we retain only the face with the largest dhkl.

Model Implementation Ž.1 Perform a standard attachment energy simulation and interplanar spacing calculation on the material of interest. The output yields the likely faces, the intermolecular bond energies and distances, and indicates which bond chains are parallel to individual crystal faces. Ž.2 Determine which bond chains correspond to stable edges with first nearest neighbors, satisfying Eq. 25. Ž.3 Refine the list of likely faces by choosing only those with at least one stable edge. For sets of forms with identical stable edges, choose the one with the larger interplanar spac- ing. Ž.4 Calculate ␥ kink for each of these bond chains using Eq. 23. Obtain ␾ kink from Eq. 15 and ␾ edge from Eqs. 27 through Figure 3. Two views of an adipic acid crystal grown from 30. aqueous solution.

2508 November 1998 Vol. 44, No. 11 AIChE Journal Table 1. Estimated Interfacial Free Energies( Crystal– Water) in the First Nearest Neighbor Bond Chains of Adipic Acid ␾ crystl ␥␥␾ cryst kink kink LabelŽ.Ž.Ž.Ž.Ž. kJrmol A˚ ergrcm22 ergrcm kJrmol U i, i 8.263 5.65 43.26 74.46 14.2 j 7.272 5.15 34.70 70.25 14.7 h 10.1 7.0 0.75 peated these calculations in order to obtain the detailed sim- ulation outputŽ. Winn, in press . The three bond chains that satisfy Eq. 25, along with a hydrogen bonded chain are re- ported in Table 1 with their magnitude and bond distance. U The bonds labeled i, i Ž.these two are related by symmetry , and j are formed primarily from dispersive forces, while h designates the hydrogen bond. The top eight likely faces from the BFDH model are listed in Table 2. For each, we have listed the bond chains that are coplanar and form stable edges. Table 1 also contains the results of the physical property predictions: ␥ cryst from the attachment energy simulation, and, using water surface properties given earlier, the calcu- Figure 4. Predicted morphology of adipic acid grown in ␥ kink ␾ kink ␥ kink lated and values. The value of h was taken water. from a measurement of interfacial free energy of water on liquid heptanoic acidŽ. Girifalco and Good, 1957 , and is very AminŽ. 1984 . The primary interest in this system is the exis- small as expected. tence of high aspect ratio needles along the b axis when grown The large kink free energy values imply the screw disloca- from a nonpolar hydrocarbonᎏusually hexane or heptaneᎏ tion mechanism. Relative growth rates were calculated, re- while equant low aspect ratio crystals are formed when grown calling that three edges form a six-sided spiral, and two edges from polar solvents such as ethanol and methanol. This was a four-sided one. For the forms with identical sets of stable discovered by researchers at the Upjohn CompanyŽ Gordon edges, we included only the one with the larger interplanar and Amin, 1984. , who patented the change in solvent as a spacing. The predicted shape is reported in Figure 4. It is in process improvement. The effect can be seen graphically in excellent agreement with the experimental shape, with a sim- Figure 5, where aspect ratio of ibuprofen crystals is plotted ilarly largeÄ4 100 form and an aspect ratio of about four. All vs. the solvent type, which is represented by its hydrogen of the predicted faces and their relative sizes closely match bonding solubility parameter. experimental results of other authorsŽ Davey et al., 1992; Klug An attachment energy calculation has been performed by and van Mil, 1994. . The success of this calculation suggests Bunyan et al.Ž. 1991 , using individual molecules as growth that the use of an estimate for the hydrogen bond interface units. We have implemented a similar simulation assuming property is validᎏthis has been confirmed with a successful ibuprofen dimer as the growth unitŽ. Winn, in press , and will prediction of water grown succinic acid crystal shapeŽ Winn, use these results as input to the solution-growth model. The in press. .

Example: Ibuprofen from polar and nonpolar sol©ents The importance of ibuprofen’s crystal shape to its process- ing and product quality has been discussed by Gordon and

Table 2. Major Faces of Adipic Acid Ranked by Inter- U planar Spacing.

Face dhkl Ž.A˚ Bond Chains R rel U 100 6.920 i, i , j 1.00 102 4.768 j 3.04 202 4.685 j 111 4.513 i 3.61 011 4.126 i, h 3.69 211 3.509 i 002 3.446 j, h 2.45 302 3.354 j Figure 5. Effect of solvent type on the aspect ratio of

U Along with their stable bond chains and the scaled relative velocities cal- ibuprofen crystals culated using the spiral growth mechanism. RedrawnfromGordonandAmin1984.Ž.

AIChE Journal November 1998 Vol. 44, No. 11 2509 Table 3. Estimated Interface Properties for Ibuprofen Table 5. Predicted Relative Growth Rates on Ibuprofen Crystals in Hexane Crystals Grown from Hexane and Ethanol

cryst cryst kink kink ␾ l ␥␥␾ Bond RRrel rel 22 LabelŽ.Ž.Ž.Ž.Ž. kJrmol A˚ ergrcm ergrcm kJrmol Face dhkl Ž.A˚ Chains Ž Hexane . Ž Ethanol . U i 18.234 7.89 39.03 2.43 1.13 100 14.742 i, j, j 1.00 1.00 U j, j 15.418 6.66 27.86 0.35 0.19 011 6.324 j rough 1.96 111 6.014 j 111 5.599 j 002 5.294 i 2.75 1.13 crystal structure of ibuprofen, like most monocarboxylic acids, 102 5.255 i is an arrangement of hydrogen-bonded dimers interacting 211 5.509 j with mainly dispersive forces. It is thought that this packing 102 4.729 i stems from dimer formation in solution prior to incorpora- tion into the crystalŽ. Gavezzotti et al., 1997 . The morphological model was implemented for two repre- sentative solvents: hexane and ethanol. For hexane, ␥ solvs22 0.007Ž. from a saturated solution at 302 K . We can use this ᎏ ergrcm2 Ž. Kaelble, 1971 . For ethanol, ␥ solvs22.8 ergrcm 2 , result as a test of our method at supersaturations lower than ⌬ Ä4 and using solubility parameters from BartonŽ. 1983 , we have this measured transition, the calculated Gc for 110 should X calculated ␥ solvs16.3 ergrcm2. The results of implementing be greater than 3RT, and at higher supersaturation the op- the model are given in Tables 3, 4, and 5. posite should be true. Growth from hexane results in low kink free energies and, An attachment energy calculation of biphenyl was per- thus, we apply the 2-D nucleation model. It is noteworthy formedŽ. Winn, in press conforming to the previous biphenyl ␾ kink r modeling of Docherty and RobertsŽ.Ž 1988b . The two simula- that j is smallŽ. 0.35 kJ mol , and we might expect rough- ening onÄ4 011 depending on the supersaturation. The model tions differ in that we employed the DREIDING force field ⌬ - Ž.Mayo et al., 1990 , and assumed only dispersive interactions, predicts Gc 3RT forÄ4 011 when the relative supersatura- tion is 0.0019 or greaterŽ. assuming T s313 K . while the other that is, no coulombic forces.. The three bonds that meet the two formsÄ4 100 and Ä4 002 remain stable. Ž Note that we use stable edge criteria are listed in Table 6, and the likely faces ␥ solvs r 2 ⌬␮s RT lnŽ. 1q ␣ , where ␣ is the relative supersaturation. are listed in Table 7. Using 28.5 erg cm for toluene The relative supersaturation is normally the relative differ- Ž.Kaelble 1971 , and, assuming it is completely nonpolar, the ence in equilibrium activities or, as an approximation, equi- kink and edge free energies were calculated. librium concentrations.. We therefore expect rough, fast growth down the b axis at moderate supersaturations. Needle shapes are predicted, which is in agreement with the patent. For growth from ethanol, we apply the screw dislocation model which results in an equant shape with an aspect ratio of about two. Comparison with Figure 5Ž ethanol hydrogen bond solubility parameter is 9.5. indicates that our prediction is again in agreement with the patent. The shapes of both predictions are drawn in Figure 6. The hexane is constructed assuming that the rough growth ofÄ4 011 is an order of magni- tudeŽ. ten times the average growth rate of the two unrough- ened, slow faces. These calculated morphologies are also very similar to experimental ones reported by Bunyan et al.Ž. 1991 .

Example: biphenyl grown from toluene This solute and solvent system are of a similar chemical nature, which should result in low kink and edge free ener- gies, and, thus, 2-D nucleated growth. In addition, we might expect face roughening to occur, even for low index faces. Jetten et al.Ž. 1984 have determined experimentally that a transition from flat to rough growth occurs on theÄ4 110 faces of biphenyl in toluene when the relative supersaturation is

Table 4. Estimated Interface Properties for Ibuprofen Crystals in Ethanol Figure 6. Predicted morphology of ibuprofen crystals ␾ crystl ␥␥␾ cryst kink kink grown from hexane() top and ethanol ( bot- LabelŽ.Ž.Ž.Ž.Ž. kJrmol A˚ ergrcm22 ergrcm kJrmol tom) . i 18.234 7.89 39.03 11.37 5.31 Hexane grown shape assumes the unstable growth in the b U j, j 15.418 6.66 27.86 8.02 4.44 direction is an order of magnitude faster than the slow growth directions.

2510 November 1998 Vol. 44, No. 11 AIChE Journal Table 6. Estimated Interfacial Properties for Biphenyl Crystals in Toluene

␾ crystl ␥␥␾ cryst kink kink LabelŽ.Ž.Ž.Ž.Ž. kJrmol A˚ ergrcm22 ergrcm kJrmol U i, i 11.157 4.94 43.83 1.36 0.36 j 8.516 5.63 38.46 0.53 0.12

⌬ Two values of Gc are presented: one calculated at a rela- tive supersaturation of 0.0065, and the other at 0.007. Since 3RT is about 7.5 kJrmol, the lower supersaturation leads to a ⌬ s r flat growth onÄ4Ž 110 Gc 7.9 kJ mol . , while the higher su- persaturation results in roughened growth Ž⌬G s 7.3 c Figure 7. Predicted crystal morphology of biphenyl kJrmol. . We have calculated a precise transition at a relative grown from toluene. supersaturation of 0.0069. Thus, our criteria for kinetic ␥ edge ⌬ roughening and our method for predicting and Gc predict a roughening transition onÄ4 110 that closely matches the experimental result. In addition, the relative growth rates One of the limitations of this approach is that it applies to calculated for the flat growth scenario produce a thin lozenge cases where there are mainly dispersive forces between the shapeŽ. see Figure 7 that matches the experimental morphol- solvent and crystal at kinks and steps, or where there are ogyŽ. Jetten et al., 1984 . known to be very specific electrostatic interactionsŽ such as Bennema and co-workersŽ Jetten et al., 1984; Bennema, water᎐carboxyl group. . The method cannot handle situations 1993. have also presented a method for predicting the rough- where forces are mostly coulombicŽ. such as inorganics or ening of faces. It is based on the theory of JacksonŽ. 1958 , where there are induced electrostatic interactions between which predicts thermodynamic roughening, and can also be solvent and crystal. Nevertheless, if improved ways of predict- used to determine qualitatively the faces likely to undergo ing interfacial free energy become availableᎏfor example, kinetic roughening. For example, Jetten et al.Ž. 1984 pre- group contribution methodsᎏthen the technique may be able dicted thatÄ4 110 is a likely candidate for kinetic roughening to handle more varied situations. Another, but less important in the above system. Their method is widely viewed as the drawback is that it cannot predict the absolute or relative standard means for predicting roughening transition; the velocities of faces undergoing rough, transport limited growth. general efficacy of our technique, which for this example was However, these fast growing sections are the most susceptible more quantitatively precise, is not yet known. to mechanical abrasion and breakage, and, therefore, even a fully detailed transport model might not result in quantita- tively accurate predictions. Conclusion The technique uses simplified versions of crystal growth We have developed a model for predicting the shape of theories. Many of the assumptions, such as the insignificance organic crystals under solution growth. It requires only or isotropy of diffusionŽ. surface or bulk and steric barriers, knowledge of pure component properties: the structure and have been previously suggested by other authors, and are energy of the solid phaseᎏwhich can be readily calculated by generally considered reasonable for a first approach. Our cri- attachment energy calculationsᎏand the surface free energy terion for nearest neighbor bonds and stable edges has not of the pure solvent. The computational time is minimal, and, been made so explicitly elsewhere. It is one of the critical thus, it can be used at the early stages of process synthesis. assumptions of this technique, eliminating many edge direc- tions and faces from consideration, and is by itself a useful tool for determining likely crystal faces. The validity of this assumption will be determined through further application of Table 7. Predicted ⌬G and Relative Growth Rates c the model. for Biphenyl Crystals in Toluene at Two Super- U saturations: ␣ s0.0065; ␣ s0.007 Another significant aspect of the model is its potential abil- 12 ity to predict the effects of certain types of additivesᎏsurface ⌬ Ž.␣ ⌬ Ž.␣ dhklBond G c 1 Gc 2 active agentsᎏon crystal shape. These materials, which are Ž.˚ Ž.Ž.Ž.Ž.r ␣ r ␣ Face A Chains kJ mol Rrel 1kJ mol R rel 2 very effective even in small amounts, are increasingly being U 001 9.472 i, i , j 66.4 1.0 61.6 1.0 explored as a means to achieve desired morphological prop- 110 4.621 i 7.9 8.4 7.3 rough erties. For example, the E. I. DuPont de Nemours Corpora- 111 4.239 i tionŽ. Klug and van Mil, 1994 have patented a process im- 111 4.072 i 200 4.044 j 0.9 rough 0.85 rough provement where they have added a surfactantŽ sodium do- 201 3.845 j decyl benzyl sulphonate. to the adipic acid process which re- 201 3.605 j sults in needle growth along the a axis.Ž This improves flowa- 112 3.395 j bility of the suspension by reducing the area of the hy- U Ä4 . Note the transition to rough growth onÄ4 110 with the increased super- drophilic 100 faces . This surfactant is known to create a saturation. hydrocarbon rich surface at the airrwater interface, which re-

AIChE Journal November 1998 Vol. 44, No. 11 2511 duces the surface free energy from 70 to 40 ergrcm2 at room Cabrera, N., and M. Levine, ‘‘On the Dislocation Theory of Evapo- temperatureŽŽ. Porter, 1991 . If one presupposes that an ration of Crystals,’’ Phil. Mag., 1, 450Ž. 1956 . Carter, P. W., A. C. Hillier, and M. D. Ward, ‘‘Nanoscale Surface equivalent hydrocarbon rich interface forms between the sol- Topography and Growth of Molecular Crystals: The Role of vent and adipic acid during crystallization, then our model Anisotropic Intermolecular Bonding,’’ J. Amer. Chem. Soc., 116, would suggest that bond chains with dispersive forces have 944Ž. 1994 . small kink free energies, and that only along hydrogen bond Chernov, A. A., Modern Crystallography III, Springer-Verlag, New Ž. chains might there be a large ␾ kink. Thus, flat slow growth York 1984 . Chernov, A. A., and T. Nishinaga, ‘‘Growth Shapes and Their Stabil- should occur only on theÄ4 011 and Ä4 002 faces, while rough ity at Anisotropic Interface Kinetics: Theoretical Aspects for Solu- and fast growth should occur along the a axis: exactly what is tion Growth,’’ Morphology of Crystals, Part A, I. Sunagawa, ed., described in the patent. This example suggests that the effect Terra Scientific, TokyoŽ. 1987 . ᎐ of surface active additives or impurities may be accounted for Clydesdale, G., R. Docherty, and K. J. Roberts, ‘‘HABIT A Pro- gram for Predicting the Morphology of Molecular Crystals,’’ Comp. using this model, when only the properties of the agent in the Phys. Comm., 64, 311Ž. 1991 . pure solvent are known. Davey, R. J., ‘‘Looking Into Crystal Chemistry,’’ The Chemical Engi- neer Ž.Dec. 1991 . Davey, R. J., S. N. Black, D. Logan, S. J. Maginn, J. E. Fairbrother, Acknowledgments and D. J. W. Grant, ‘‘Structural and Kinetic Features of Crystal Growth Inhibition: Adipic Acid Growing in the Presence of n-Al- This research was supported by the University of Massachusetts kanoic Acids,’’ J. Chem. Soc. Faraday Trans., 88, 3461Ž. 1992 . Process Design and Control Center. Special thanks to Professor Klaus Docherty, R., and K. J. Roberts, ‘‘MORANG᎐A Computer Program Wintermantel and Drs. Matthias Kind, Simon Jones, and Matthias Designed to Aid in the Determination of Crystal Morphology,’’ Rauls, who made it possible for one of usŽ. D.W. to perform the Comp. Phys. Comm., 51, 423Ž. 1988a . crystallization experiments in the laboratories of BASF AG. Docherty, R. and K. J. Roberts, ‘‘Modelling the Morphology of Molecular Crystals; Application to Anthracene, Biphenyl and ␤- Succinic Acid,’’ J. Crystal Growth, 88, 159Ž. 1988b . Notation Donnay, J. D. H., and D. Harker, ‘‘A New Law of Crystal Morphol- max s Ž. Ah, k, l reticular area of the most dense face intersecting h-bond ogy Extending the Law of Bravais,’’ Amer. Min., 22, 446 1937 . chains, A˚2rmolecule Freidel, M. G., ‘‘Etudes´ sur la loi de Bravais,’’ Bull. Soc. Franc. Miner., a, b, cscrystallographic axes 9, 326Ž. 1907 . Ž.h, k, l scrystallographic plane Ž denotes a crystal face . Gavezzotti, A., G. Filippini, J. Kroon, B. P. van Eijck, and P. Klew- Ä4h, k, l scrystal form Ž family of symmetrically related planes . inghaus, ‘‘The Crystal Polymorphism of Tetrolic Acid: A Molecu- wxh, k, l scrystallographic direction lar Dynamics Study of Precursors in Solution, and Crystal Struc- h, i, jslabels of edge directionsŽ. bond chains ture Generation,’’ Chem. Eur. J., 3, 893Ž. 1997 . s Dm molecular diameterŽ. A˚ Giacovazzo, C., ‘‘Symmetry in Crystals,’’ Fundamentals of Crystallog- ⌬ s Ž. Gc free energy to form a critical 2-D mucleus, kJ or a mole of raphy, C. Giacovazzo, ed., Oxford Univ. Press, Oxford 1992 . critical nuclei, kJrmol Gilmer, G. H., and P. Bennema, ‘‘Simulation of Crystal Growth with r sradius of a 2-D nucleusŽ. A˚ Surface Diffusion,’’ J. Appl. Phys., 43, 1347Ž. 1972 . s rc radius of a 2-D nucleus of critical sizeŽ. A˚ Girifalco, L. A., and R. J. Good, ‘‘A Theory for the Estimation of Rsgas constant, kJrmolؒK Surface and Interfacial Energies: I. Derivation and Application to T stemperature, K Interfacial Tension,’’ J. Phys. Chem., 61, 904Ž. 1957 . s 3 Vm molecular volume, A˚ Gordon, R. E., and S. I. Amin, ‘‘Crystallization of Ibuprofen,’’ U. S. ysdistance between steps on a crystal face, A˚ Patent Number 4,476,248Ž. 1984 . ␮schemical potential of a solute in solution, kJrmol Grant, D. J. W., K. Y. Chow, and H. Chan, ‘‘Modification of the ␮sat schemical potential of a solute in a saturated solution, Physical Properties of Adipic Acid by Crystallization in the Pres- kJrmol ence of n-Alkanoic Acids,’’ Particle Design Via Crystallization,R. sat ⌬␮ ssupersaturation: ⌬␮s␮y␮ ,kJrmol Ramanarayanan, W. Kern, M. Larson and S. Sikdar, eds., AIChE ␾ edge swork to create an edge on a crystal face, kJrmol Ž.1991 . ␾ kink swork to create a kink in a edge, kJrmol Green, D. A., P. Meenan, G. M. Sunshine, and B. J. Enis, ‘‘Habit Modelling and Verification for Glutaric Acid,’’ J. Phys. D., 26, B35 Ž.1993 . Literature Cited Hansen, C., and A. Beerbower, ‘‘Solubility Parameters,’’ Kirk-Othmer Encyclopedia of Chemical Technology, Wiley, New YorkŽ. 1971 . Adamson, A. W., Physical Chemistry of Surfaces, Wiley, New York Hartman, P., and W. G. Perdok, ‘‘On the Relations Between Struc- Ž.1990 . ture and Morphology of Crystals: I,’’ Acta. Cryst., 8, 49Ž. 1995 . Barton, A. F. M., Handbook of Solubility and Other Cohesi®e Parame- Hildebrand, J. H., and R. L. Scott, Regular Solutions, Prentice-Hall, ters, CRC Press, Boca Raton, FLŽ. 1983 . Englewood Cliffs, NJŽ. 1962 . Bennema, P., ‘‘Growth and Morphology of Crystals,’’ Handbook of Hoffman, D. W., and J. W. Cahn, ‘‘A Vector Thermodynamics for Crystal Growth 1, D. T. J. Hurle, ed., North-Holland, New York Anisotropic Surfaces,’’ Surf. Sci., 31, 389Ž. 1972 . Ž.1993 . Hunan, H. J., J. P. van der Eerden, L. A. M. J. Jetten, and J. G. M. Berkovitch-Yellin, Z., ‘‘Toward an Ab Initio Derivation of Crystal Odekerken, ‘‘On the Roughening Transition of Biphenyl: Faceted Morphology,’’ J. Amer. Chem. Soc., 107, 8239Ž. 1985 . to Non-Faceted Growth of Biphenyl for Growth from Different Boek, E. S., W. J. Briels, and D. Feil, ‘‘Interfaces between Saturated Solvents and the Melt,’’ J. Crystal Growth, 51, 589Ž. 1981 . Aqueous Urea and Crystalline Urea: A Molecular Dynamics Jackson, K. A., ‘‘Mechanism of Growth,’’ Liquid Metals and Solidifica- Study,’’ J. Phys. Chem., 98, 1674Ž. 1994 . tion, Amer. Soc. for Metals, ClevelandŽ. 1958 . Bravais, A., Etudes´ Crystallrographiques, Gauthier-Villars, Paris Jetten, L. A. M. J., H. J. Hunan, P. Bennema, and J. P. van der Ž.1866 . Eerden, ‘‘On the Observation of the Roughening Transition of Or- Bunyan, J. M. E., N. Shankland, and D. B. Sheen, ‘‘Solvent Effects ganic Crystals Growing from Solution,’’ J. Crystal Growth, 68, 503 on the Morphology of Ibuprofen,’’ Particle Design Via Crystalliza- Ž.1984 . tion, Ramanarayanan, R. W. Kern, M. Larson and S. Sikdar, eds., Kaelble, D. H., Physical Chemistry of Adhesion, Wiley-Interscience, Amer. Inst. of Chem. Engs., New YorkŽ. 1991 . New YorkŽ. 1971 . Burton, W. K., N. Cabrera, and F. C. Frank, ‘‘The Growth of Crys- Kaichew, R., ‘‘The Molecular-Kinetic Theory of Crystal Growth,’’ tals and the Equilibrium Structure of their Surfaces,’’ Phil. Trans. Growth of Crystals, Vol. 3, A. V. Shubnikov and N. N. Sheftal, eds., Roy. Soc., A243, 299Ž. 1951 . Consultants Bureau, New YorkŽ. 1962 .

2512 November 1998 Vol. 44, No. 11 AIChE Journal Kitaigorodsky, A. I., Molecular Crystals and Molecules, Academic the 2-D nucleation model when the free energy of kink for- Press, New YorkŽ. 1973 . mation is low, there is likely to be a high density of kinks on Klug, D. L., and J. H. van Mil, ‘‘Adipic Acid Purification,’’ United States Patent Number 5,296,639Ž. 1994 . any edge: there is a high probability that molecules entering Lewis, B., ‘‘The Growth of Crystals of Low Supersaturation: I. The- the solid state form kinks. The amount of desolvation of these ory,’’ J. Crystal Growth, 21, 29Ž. 1974 . molecules can be estimated using the proportionality approxi- Liu, X. Y., and P. Bennema, ‘‘The Relationship Between Macro- mation Ž.Liu and Bennema, 1996 , which assumes that the scopic Quantities and the Solid-Fluid Interfacial Structure,’’ J. ᎐ Ž. anisotropy of solute solvent bonding is proportional to the Chem. Phys., 98, 5863 1993 . ᎐ Liu, X. Y., E. S. Boek, W. J. Briels, and P. Bennema, ‘‘Prediction of anisotropy of solid solid bonding. It can be expressed as Crystal Growth Morphology Based on Structural Analysis of the Solid-Fluid Interface,’’ Nature, 374, 342Ž. 1995 . U sl Esl Liu, X. Y., and P. Bennema, ‘‘Growth Morphology of Crystals and s Ž.34 the Influence of Fluid Phase,’’ Crystal Growth of Organic Materials, UEtot latt A. S. Myerson, D. A. Green and P. Meenan, eds., ACS, Washing- tonŽ. 1996 . and Manne, S., J. P. Cleveland, G. D. Stucky, and P. K. Hansma, ‘‘Lattice Resolution and Solution Kinetics on Surfaces of Amino Acid Crys- att att tals: An Atomic Force Microscopy Study,’’ J. Crystal Growth, 130, U E Ž. s Ž.35 133 1993 . UEtot latt Markov, I. V., Crystal Growth for Beginners, World Scientific, Singa- poreŽ. 1995 . Mayo, S. L., B. D. Olafson, and W. A. Goddard, III, ‘‘DREIDING: where U sl is the amount of energyŽ. kJrmol to desolvate the A Generic Force Field for Molecular Simulations,’’ J. Phys. Chem., portion of the molecule that is parallel to the crystal face Ž. 94, 8897 1990 . Ž.lateral interactions , and U att is the amount to desolvate the Muller-Krumbhaar, H., ‘‘Kinetics of Crystal Growth,’’ Current Topics in Material Science, Vol. I, E. Kaldis, ed., North-Holland, Amster- molecule in the direction orthogonal to the crystal face damŽ. 1978 . Ž.kJrmol . The sum of the two are constant and equal to U tot. Myerson, A. S., and R. Ginde, ‘‘Crystals, Crystal Growth, and Nucle- The symbols Esl and Eatt are the slice and attachment ener- ation,’’ Handbook of Crystal Growth, A. S. Myerson, ed., Butter- giesŽ. kJrmol , analogous properties that refer to the Ž. worth-Heinemann, Boston 1993 . ᎐ Ž Ohara, M., and R. C. Reid, Modeling Crystal Growth Rates from Solu- solid solid bonding in the crystal see Hartman and Perdok, tion, Prentice-Hall, Englewood Cliffs, NJŽ. 1973 . 1955, for details. . Their sum is equal to the lattice energy Porter, M. R., Handbook of Surfactants, Chapman and Hall, New E latt Ž.kJrmol which is a constant. YorkŽ. 1991 . For a simple cubic crystal lattice, a molecule adsorbing onto Saska, M., and A. S. Myerson, ‘‘The Theoretical Shape of Sucrose an edge must lose one-half of its orthogonal interactions with Crystals from Solution,’’ J. Crystal Growth, 61, 546Ž. 1983 . Tanguy, D., and P. Marchal, ‘‘Relations between the Properties of the solvent and one-quarter of its lateral interactions with the Particles and their Process of Manufacture,’’ Trans. IChemE, 74, solvent. We can express the desolvation energy as 715Ž. 1996 . van der Eerden, J. P., ‘‘Crystal Growth Mechanisms,’’ Handbook of s slr q attr Crystal Growth, D. T. J. Hurle, ed., North-Holland, Amsterdam U U 4 U 2 Ž.1993 . Eslr4q Eattr2 Voronkov, V. V., ‘‘Dislocation Mechanism of Growth with a Low tot ® s U Kink Density,’’ So . Phys. Crystallog., 18, 19Ž. 1973 . ž/E latt Wells, A. F., ‘‘Crystal Habit and Internal Structure: I,’’ Phil. Mag., 37, 184Ž. 1946 . Eslr4 Winn, D., ‘‘Predicting Crystal Shape in Organic Solids Processes,’’ s 1r2y U tot Ž.36 PhD Thesis, Univ. of Massachusetts, AmherstŽ. in press . ž/E latt Yip, C. M., and M. D. Ward, ‘‘Atomic Force Microscopy of Insulin Single Crystal: Direct Visualization of Molecules and Crystal Growth,’’ Biophys. J., 71, 1071Ž. 1996 . Therefore, the greater the slice energy of a face, the smaller the desolvation energy. The total desolvation energy is ap- Appendix proximately the difference between the sublimation enthalpy tot f⌬ y⌬ and the dissolution enthalpy ŽU HsubH diss, which 2-D nucleation birth and spread model implies U tot ;10 to 100 kJrmol. , and thus U has the same ⌬ The birth and spread model of crystal growth is thought to order of magnitude as Gc for organics in solution. be the most realistic of the 2-D nucleation growth models We also know that the greater the slice energy, the greater Ž.Ohara and Reid, 1973 . It is often assumed that the most the magnitude of unsaturated interactions at the edges, and ␥ edge ⌬ significant face-dependent variables in this mechanism are the the greater the value of . Since Gc increases with the ␥ edge ⌬ activation energy for desolvation U and the free energy square of , it is expected that Gc increases more rapidly ⌬ change to form a critical nucleus Gc ŽLiu and Bennema, than U decreases. Therefore, the relative growth rates should ⌬ 1996.Ž . The latter dictates the equilibrium concentration of still decrease as Gc increases, but not at an exponential rate. critical nuclei..Ž. Using the development of Markov 1995 , we We, therefore, propose the following expression for the rela- can write the expression for relative growth rates as tive growth rate for growth by 2-D nucleation

A y r y⌬ r A r⌬ Rrel exp Ž.U RT exp Ž.Gc 3RT Ž.33 Rrel 1 Gc Ž.37

Molecules attaching to nuclei can either enter into kinks, Equation 37 simplifies the growth model of Eq. 33 and Eq. or adsorb onto flat edges so as to form kinks. Since we apply 36 and provides good morphological predictions.

AIChE Journal November 1998 Vol. 44, No. 11 2513 Spiral growth model desolvation energy In the spiral growth model, the desolvation step is also part Ž.Eattr2q Eslr2 of the complete growth mechanism. However, the spiral Us U tot mechanism is applied when kink free energies are high and ž/E latt kink densities are low, and, thus, most solute molecules are s totr assumed to be incorporating into kink sitesŽ as opposed to U 238Ž. making kink sites. . A molecule entering into a kink position Therefore, the desolvation energy for entering a kink is forms, on average, half of its total solid᎐solid bonds: half of isotropic, and we need not include the desolvation step in the the orthogonal interactions, and half of the lateral interac- relative growth rate expressions for spiral growth. tions.Ž. In the simple cubic lattice, this is exactly true . Using the proportionality approximation, we can again estimate the Manuscript recei®ed May 18, 1998, and re®ision recei®edSept.3,1998.

2514 November 1998 Vol. 44, No. 11 AIChE Journal